Scott Hughes
12 April 2005
Massachusetts Institute of Technology
Department of Physics
8.022 Spring 2005
Lecture 16:
RL Circuits.
Undriven RLC circuits; phasor representation.
16.1
RL circuits
Now that we know all about inductance, it’s time to start thinking about how to use it in
circuits. A simple circuit that illustrates the major concepts of inductive circuits is this:
V
L
R
S2
S1
Suppose that at
t
= 0, we close the switch
S
1, leaving
S
2 open. How does the current evolve
in the circuit after this?
Let’s first think about this physically. Lenz’s law tells us that the magnetic flux through
the inductor does not “want” to change. Since this flux is initially zero, the inductor will
impede any current that tries to flow it — the current will initially try to remain at zero.
As time passes, current will gradually leak through, and the magnetic flux will build up.
Eventually, it should saturate at
I
=
V/R
— at this point the most current possible is
flowing through the circuit.
16.1.1
To Kirchhoff or not to Kirchhoff
We’d now like to analyze this circuit quantitively as well.
In all other circumstances in
which we’ve examined a circuit, we’ve used Kirchhoff’s laws for this; for a a simple single
loop circuit, the relevant rule is Kirchhoff’s second law, “The sum of the EMFs and the
voltage drops around a closed loops is zero”.
This rule is essentially a restatement of the old electrostatics rule that
H
~
E
·
d~s
= 0.
This
rule DOES NOT HOLD when we have a time changing magnetic field!
In other
words, inductors completely invalidate — at least formally — the foundation of Kirchhoff’s
second law.
146
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So what do we do? Let’s reconsider: in place of
H
~
E
·
d~s
= 0, we have Faraday’s law:
I
~
E
·
d~s
=

1
c
d
Φ
B
dt
=
E
ind
=
L
dI
dt
.
The closed loop integral of
~
E
gives us the induced EMF. I personally find it best to think of
the inductor as an EMF source; it acts in the circuit effectively like a battery would. Even
though Kirchhoff’s laws no longer stand, formally, on a very solid foundation, we can extend
the formalism by thinking of inductors in the circuit as a source of EMF. The equations
then carry over just fine; indeed, they carry over so well that almost all textbooks tell you
to just apply Kirchhoff’s laws to circuits containing inductors without even mentioning this
subtlety. It’s always worth bearing in mind that what you are
really
using is Faraday’s law.
I will try to refer to the loop rule as Faraday’s law; it’s such a habit to call it Kirchhoff’s
law, though, that I will surely slip from time to time.
The tricky thing in these kinds of problems is getting the signs right. In
all
of these cases,
Lenz’s law should guide you: the induced EMF will act in such a way as to oppose changes
in the inductor’s magnetic flux. If a circuit initially has no current flowing through it, the
inductor will oppose the buildup of current; if it initially
does
have current flowing, the
inductor will try to keep that current flowing. Keep this intuition in mind and you should
have
no
problem choosing the correct sign for the inductor’s EMF.
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 Spring '10
 ScottHughes
 Inductor, RL circuit, LC circuit, Lenz

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